Classical mechanics · Visual essay

Gravity as wavefronts of action

A thrown particle traces a parabola. A whole family of throws can be drawn as a wave: a surface that spreads, bends and folds. Hamilton’s action connects the two pictures.

What are these surfaces, and why do the particles follow their normals?

A glimpse of OpticalGravity. We will build up to this picture from motion in uniform gravity.

The action of a path

For a path $𝐫 (t)$ , the action is the time integral of kinetic energy minus potential energy:

𝒜 [𝐫] = \int_{t_{A}}^{t_{B}} (\frac{m}{2} {| \dot{𝐫} |}^{2} - V (𝐫)) d t

A physical path makes this integral stationary under small changes of the path with its endpoints and endpoint times fixed. “Stationary” means that the first-order change vanishes; it need not always be a minimum. The resulting equation of motion is Newton’s law.

From fixed endpoints to free endpoints

Keep the initial position and time fixed. For each possible arrival event $(𝐫, t)$ , evaluate the action on a physical path connecting the two. The result is Hamilton’s principal function, $S (𝐫, t) = 𝒜 [𝐫_{cl}]$ . We have restricted the path functional to solutions of Newton’s law; its value is now a function of where and when the particle arrives. If several paths are possible, work on one smooth branch at a time.

Move the arrival point slightly, keeping its time fixed. In the first variation, the interior term vanishes by the equations of motion. The boundary term from integration by parts remains:

d S = 𝐩 \cdot d 𝐫 ⟹ \nabla S = 𝐩 = m 𝐯 .

Here “free” means comparing paths with different endpoints; this change in action is generally nonzero. Allowing the arrival time to vary as well gives

d S = 𝐩 \cdot d 𝐫 - H d t, \frac{\partial S}{\partial t} = - H .

Here $H$ is the total energy, kinetic plus potential; the partial time derivative holds the arrival position fixed. Replacing the momentum in that energy by $\nabla S$ gives

\frac{\partial S}{\partial t} + \frac{{| \nabla S |}^{2}}{2 m} + V = 0 .

This is the Hamilton–Jacobi equation: an equation for an action field over space and time. Its level surfaces have normals along the particle momentum. A precise connection to wave phase comes from writing a matter wave as an amplitude times a phase factor:

ψ = a e^{i S / ℏ} .

Here $a$ is the amplitude, and $S / ℏ$ is the phase angle; the reduced Planck constant $ℏ$ converts action into a dimensionless phase. Equal values of $S$ therefore describe equal-phase surfaces. The local wavevector is $𝐤 = \nabla S / ℏ = 𝐩 / ℏ$ : momentum fixes the direction of the phase normal and how closely the fronts are spaced.

Substituting this form into Schrödinger’s equation gives the Hamilton–Jacobi equation above with one additional term involving the spatial curvature of the amplitude. Where that term is negligible, classical action supplies the wave’s phase. This is the semiclassical approximation, closely related to geometrical optics. Near a caustic it can fail; diffraction and the wave amplitude need a fuller wave calculation.

Abbreviated Action W

Suppose the potential depends on position but not time. Energy is then conserved along each trajectory, and we can select a family sharing one value $E$ . We are now choosing paths by their energy rather than prescribing a common travel time.

For the corresponding fixed-energy solution, $\frac{\partial S}{\partial t} = - E$ . Integrating in time gives $S = W - E t$ . The spatial part $W$ is Hamilton’s characteristic function, or abbreviated action. It retains the momentum gradient, so its increment along a route is momentum dotted with displacement. Measured from departure at the source $A$ ,

S_{j} = W_{j} - E t, W_{j} (𝐫; E) = \int_{A}^{𝐫} 𝐩 \cdot d 𝐪 .

The index $j$ distinguishes different routes to the same point. Each route carries its own action, giving a separate smooth branch of the function.

Same launch speed. Same destination. Different travel times. Both particles leave A together at 6 m/s. The cyan particle reaches B after 0.49 s and waits there; the violet particle arrives after 1.12 s. Recorded from Action as Wave in uniform gravity.

A moving front is a level surface $S_{j} = C$ . Equivalently,

W_{j} (𝐫; E) = E t + C .

One value of $C$ gives one front; several equally spaced values give a train of fronts. The clock $t$ advances the whole pattern. It is distinct from the travel time along any particular trajectory.

In this fixed-energy form, the momentum relation and the Hamilton–Jacobi equation become

𝐩 = \nabla W_{j}, {| \nabla W_{j} |}^{2} = 2 m (E - V) .

The gradient is perpendicular to a level surface. Since momentum points along the particle’s velocity, the trajectories cross their action fronts at right angles. They are the rays of this wave picture.

Gravity

Now take a uniform gravitational field pointing down. With height $z$ measured upwards and the source at $z = 0$ , the potential is $V = m g z$ . Launching every particle with the same speed $v_{0}$ selects the common energy $E = \frac{1}{2} m v_{0}^{2}$ .

Let the launch direction vary freely. Some particles rise, some fall immediately, and each follows an ordinary projectile trajectory. The upward throws touch a common boundary: the violet curve in the first clip below. Rotate this picture around its vertical axis and the boundary becomes a surface. It is the caustic of this family of trajectories.

01Equal launch speeds, different directions. The violet envelope marks the limit of reach.

02A surface of equal action. The camera pauses at a side view, then returns to show the depth of the fold.

The second clip shows the action-front construction in this field. Along each trajectory, find the point where the abbreviated action reaches the chosen value. Joining those points gives a front; advancing the phase clock moves it. A typical reachable point has two routes from the source—a low, quick route and a higher, slower one—so the action has two branches.

Changing the launch speed changes $E$ , so it changes the equation for $W$ and the shapes of its fronts. A faster launch also raises the caustic and widens the reachable region.

03A continuous family of action fronts. The open section exposes both sheets; the camera turn reveals the shape of the surfaces.

04The particles enter with their trajectories. Follow a bead through the surfaces: the trajectory is a ray, normal to its own action sheet.

There is an important distinction here. Particles launched together have the same flight time. At each instant they lie on a sphere whose centre falls under gravity. An action front instead joins points with the same action; those points can have different flight times. This is why the beads move through the surfaces.

At the fold

Follow one of the upward trajectories. It rises, turns and begins to fall. A little later, it touches the caustic tangentially. That contact is where the action sheet changes branch. Except for the straight-up throw, the contact occurs after the particle’s highest point.

At the caustic, the two routes to a point merge. The front folds there, and the particle continues smoothly onto the other sheet. The color change marks that transition.

05Twenty-eight trajectories meet the fold. Launch times are staggered so that each bead reaches its caustic contact as the moving edge arrives. The curved label follows that contact along the parabola.

Follow the change

Apricot
Before the trajectory touches the caustic.

The bright edge
The sheets meet at the caustic contact.

Violet
The trajectory continues beyond its tangency, on the other action sheet.

The launch speed and gravitational field are the same for every bead. Their launch times are chosen to make the contact visible; they cross the front freely as they move.

In a vertical section, the caustic is the parabola

z_{c} (ρ) = \frac{v_{0}^{2}}{2 g} - \frac{g ρ^{2}}{2 v_{0}^{2}},

where $ρ$ is horizontal distance from the source axis. Every upward ray touches this boundary from inside. The boundary is the envelope of the trajectories, and the fold of the action fronts is another way to see the same geometry.

How the gravity fronts are calculated

Let $T$ denote elapsed flight time and let $θ$ be the launch angle from the upward vertical. Write $c = \cos θ$ . The projectile follows

\begin{array}{l} ρ = v_{0} \sqrt{1 - c^{2}} T, \\ z = v_{0} c T - \frac{g}{2} T^{2} . \end{array}

Along that trajectory, $𝐩 \cdot d 𝐪 = m {| 𝐯 |}^{2} d T$ . Integrating gives

W (c, T) = m (v_{0}^{2} T - v_{0} g c T^{2} + \frac{g^{2}}{3} T^{3}) .

For each direction, solve $W (c, T) = E t + C$ for $T$ , then put that time into the projectile equations. Those points form the front. Revolving the meridional curve gives the three-dimensional surface.

For an upward launch, the caustic contact occurs at $T_{c} = \frac{v_{0}}{g c}$ . The top of the individual arc occurs earlier, at $T_{top} = \frac{v_{0} c}{g}$ . This difference is why turning downwards and changing color are separate events.

Where the initial speed enters Hamilton’s principal function

There is another useful form of the principal function. Fix the source at the origin, the endpoint at $(ρ, z)$ , and the flight duration $T$ . The action evaluated along that classical path is

S_{cl} = \frac{m (ρ^{2} + z^{2})}{2 T} - \frac{m g z T}{2} - \frac{m g^{2} T^{3}}{24} .

No launch speed appears explicitly: the endpoint and duration already determine the required initial velocity. To pick out our chosen speed, impose the energy condition

- \frac{\partial S_{cl}}{\partial T} = E = \frac{1}{2} m v_{0}^{2} .

At a typical reachable endpoint this selects two durations, $T_{j}$ . Each gives an abbreviated-action branch

W_{j} (𝐫; E) = S_{cl} (𝐫, T_{j}) + E T_{j} .

Finally use $S_{j} (𝐫, t) = W_{j} (𝐫; E) - E t$ to move the fronts. The selected travel duration $T_{j}$ and the running phase clock $t$ have different jobs. Replacing one by the other would describe a different construction.

From throws to orbits

Now put an attracting mass above the source. The force points towards it and grows as the inverse square of distance. Choose a launch speed below escape speed, and the trajectories become Kepler ellipses.

The construction of the fronts is the same. Follow each orbit, accumulate its action and join points with equal values. The central mass is a focus of each ellipse. Different launch directions give different angular momenta, different close approaches and different paths through space.

06From OpticalGravity. The attracting mass is marked

M

above the source. The pale outer ellipse is a section of the caustic; apricot and violet retain the same branch meaning as in uniform gravity.

The surfaces bend around the mass and fold at the caustic. The particle trajectories touch the same envelope, then continue on the other action sheet. After a full orbit, they return to their starting point.

Look near the mass, where the particles move fastest. The fronts are more closely spaced there: the gradient of action is larger, so a fixed change in action takes a shorter distance. This is the same momentum–gradient relation we used for the falling particles.

A particle can complete its orbit while a front is still making its way through the pattern. The speed of a phase surface and the speed of a particle are different quantities. Their geometric connection is the direction of motion: each ray follows the normal to its own smooth action sheet.

Both animations use zero potential energy at the source. Choosing another energy zero leaves the trajectories unchanged, while changing how a fixed phase value moves in time.

Why it looks like optics

In optics, the same construction has a familiar name: the eikonal equation. Write the phase of a monochromatic wave as $k_{0} χ (𝐫) - ω t$ , where $k_{0} = ω / c$ is the vacuum wavenumber. The eikonal $χ$ has units of length. In a smooth, isotropic medium with refractive index $n$ , the short-wavelength limit of the wave equation gives

{| \nabla χ |}^{2} = n^{2} (𝐫) .

At a fixed time, constant $χ$ surfaces are phase fronts, and the rays follow their normals. Compare this with the fixed-energy Hamilton–Jacobi equation:

{| \nabla W |}^{2} = 2 m (E - V) = p^{2} .

Here $p$ is the momentum magnitude. Use the launch momentum $p_{0} = m v_{0}$ as a constant scale: $W / p_{0}$ then has units of length and obeys the optical equation with an effective index $n_{eff} = p / p_{0}$ . Higher momentum gives more closely spaced action fronts, just as a higher refractive index gives more closely spaced optical fronts.

The optical path accumulates refractive index along a ray, just as $W$ accumulates momentum. Fermat’s principle and stationary abbreviated action express the same ray geometry. In both pictures, a caustic is an envelope of rays and a fold in their phase surfaces.

The colors label the action branches. The glow and transparency help us see the geometry; they do not represent a calculated wave amplitude. Interference, diffraction and the intensity near a caustic would require a further wave calculation.

For the Hamilton–Jacobi equation, its fixed-energy form and the connection to wave mechanics, see David Tong, Classical Dynamics, sections 4.7–4.8.1. For the eikonal equation and optical path, see Ivan Bazarov, Geometric optics, page 2.